Following a recent paper by Alday and Tachikawa, we compute the instanton partition function in the presence of the surface operator by the localization formula on the moduli space. For $SU(2)$ theories, we find an exact agreement with conformal field theory (CFT) correlation functions with a degenerate operator insertion, which enables us to work out the decoupling limit of the superconformal theory with four flavors to asymptotically free theories at the level of differential equations for CFT correlation functions (irregular conformal blocks). We also argue that the $K$ theory (or five-dimensional) lift of these computations gives open topological string amplitudes on local Hirzebruch surface and its blow ups, which is regarded as a geometric engineering of the surface operator. By computing the amplitudes in both A and B models we collect convincing evidences of the agreement of the instanton partition function with surface operator and the partition function of open topological string.

We extend the direct integration method of the holomorphic anomalyequations to general $\Omega$ backgrounds $\epsilon_1\neq -\epsilon_2$for pure SU(2) $N=2$ Super-Yang-Mills theory and topological stringtheory on non-compact Calabi-Yau threefolds. We find that anextension of the holomorphic anomaly equation, modularity and boundary conditions provided by the perturbative terms as well as by the gap condition at the conifold are sufficient to solve the generalized theory in the above cases. In particular, we use the method to solve the topological string for the general $\Omega$ backgrounds on non-compact toric Calabi-Yau spaces. The conifold boundary condition follows from that the $N=2$ Schwinger-loop calculation with Bogomol'nyi-Prasad-Sommerfield (BPS) states coupled to a self-dual and an anti-self-dual field strength. We calculate such BPS states also for the large base decompactification limit of Calabi-Yau spaces with regular $K3$ fibrations and half $K3$s embedded in Calabi-Yau backgrounds.

Let $M\to N$ (resp. $C\to N$) be the fibre bundle of pseudo-Riemannian metrics of a given signature (resp. the bundle of linear connections) on an orientable connected manifold $N$. A geometrically defined class of first-order Ehresmann connections on the product fibre bundle $M\times_NC$ is determined such that, for every connection $\gamma$ belonging to this class and every $\operatorname{Diff}N$-invariant Lagrangian density $\Lambda $ on $J^1(M\times _NC)$, the corresponding covariant Hamiltonian $\Lambda ^\gamma $ is also $\operatorname{Diff}N$-invariant. The case of $\operatorname{Diff}N$-invariant second-order Lagrangian densities on $J^2M$ is also studied and the results obtained are then applied to Palatini and Einstein-Hilbert Lagrangians.

In this paper, we investigate quotients of Calabi-Yau manifolds $Y$ embedded in Fano varieties $X$, which are products of two del Pezzo surfaces — with respect to groups $G$ that act freely on $Y$. In particular, we revisit some known examples and we obtain some new Calabi-Yau varieties with small Hodge numbers. The groups $G$ are subgroups of the automorphism groups of $X$, which is described in terms of the automorphism group of the two del Pezzo surfaces.

It has been argued based on electric–magnetic duality and other ingredients that the Jones polynomial of a knot in three dimensions can be computed by counting the solutions of certain gauge theory equations in four dimensions. Here, we attempt to verify this directly by analyzing the equations and counting their solutions, without reference to any quantum dualities. After suitably perturbing the equations to make their behavior more generic, we are able to get a fairly clear understanding of how the Jones polynomial emerges. The main ingredient in the argument is a link between the four-dimensional gauge theory equations in question and conformal blocks for degenerate representations of the Virasoro algebra in two dimensions. Along the way we get a better understanding of how our subject is related to a variety of new and old topics in mathematical physics, ranging from the Bethe ansatz for the Gaudin spin chain to the M-theory description of Bogomol’nyi-Prasad-Sommerfield (BPS) monopoles and the relation between Chern–Simons gauge theory and Virasoro conformal blocks.